#128930

order	512
nilpotency class	4
isoclinism	isoclinic class of #11989
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

373 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2,4,3)(5,6)(7,17,20,40)(8,13)(9,27)(10,14)(11,31)(12,19,33,22)(15,30)(16,28)(18,35,44,57)(21,34,48,60)(23,32,46,58)(24,37)(25,54)(26,55)(29,41)(36,43)(38,53)(39,50)(42,49,56,47)(45,59,65,68)(51,67,66,61)(52,63)(62,64);(5,7,9,21,28,22,50,49,10,20,29,48,11,19,30,47)(6,12,15,34,31,17,41,42,14,33,39,60,16,40,27,56)(8,23,25,51,53,44,64,65,24,46,52,66,26,18,43,45)(13,35,36,61,55,58,63,68,37,57,62,67,38,32,54,59);(1,3,4,2)(5,8,10,24)(6,13,14,37)(7,18)(9,25,29,52)(11,26,28,53)(12,32)(15,36,39,62)(16,38,31,55)(17,35)(19,46)(20,44)(21,45)(22,23)(27,54,41,63)(30,43,50,64)(33,58)(34,59)(40,57)(42,61)(47,66)(48,65)(49,51)(56,67)(60,68)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ei,d^2=gi,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36126);